Seminar: Abel-Jacobi Map For Cubic Threefold

Seminar| Institute of Mathematical Sciences

Abstract: Abel-Jacobi map for a compact Riemann surface relates points on the Riemann surface to its Jacobian variety. It arises from the study of elliptic integrals. This is generalized by Griffiths in higher dimensions. For example, for a cubic threefold, the Abel-Jacobi map sends a pair of lines to the intermediate Jacobian. It is given by the integral of a closed (2,1) against a 3-chain bounding the two lines, modulo the periods. Recently, we proved that the Abel-Jacobi map for cubic threefold extends to a double cover of a component of the Hilbert scheme. This is used to compactify the parameter space of vanishing cycles on hyperplane sections of the cubic threefold. In this talk, I will explain these results, and introduce some related questions for cubic fourfolds.